Question: $\dfrac{ 9v + 3w }{ 7 } = \dfrac{ v - 2x }{ -8 }$ Solve for $v$.
Multiply both sides by the left denominator. $\dfrac{ 9v + 3w }{ {7} } = \dfrac{ v - 2x }{ -8 }$ ${7} \cdot \dfrac{ 9v + 3w }{ {7} } = {7} \cdot \dfrac{ v - 2x }{ -8 }$ $9v + 3w = {7} \cdot \dfrac { v - 2x }{ -8 }$ Multiply both sides by the right denominator. $9v + 3w = 7 \cdot \dfrac{ v - 2x }{ -{8} }$ $-{8} \cdot \left( 9v + 3w \right) = -{8} \cdot 7 \cdot \dfrac{ v - 2x }{ -{8} }$ $-{8} \cdot \left( 9v + 3w \right) = 7 \cdot \left( v - 2x \right)$ Distribute both sides $-{8} \cdot \left( 9v + 3w \right) = {7} \cdot \left( v - 2x \right)$ $-{72}v - {24}w = {7}v - {14}x$ Combine $v$ terms on the left. $-{72v} - 24w = {7v} - 14x$ $-{79v} - 24w = -14x$ Move the $w$ term to the right. $-79v - {24w} = -14x$ $-79v = -14x + {24w}$ Isolate $v$ by dividing both sides by its coefficient. $-{79}v = -14x + 24w$ $v = \dfrac{ -14x + 24w }{ -{79} }$ Swap signs so the denominator isn't negative. $v = \dfrac{ {14}x - {24}w }{ {79} }$